Homework 3.6 piecewise functions answers unlock the door to a mathematical realm where functions take on different forms over distinct intervals. This guide delves into the intricacies of piecewise functions, providing a clear understanding of their concept, notation, and applications in the real world.

Within this comprehensive resource, you will find meticulously organized answers to Homework 3.6, presented in an easy-to-read HTML table format. Each problem is accompanied by a detailed solution and a corresponding graph, ensuring a thorough understanding of the underlying concepts.

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over different intervals of the independent variable. They are commonly used to model situations where the relationship between the independent and dependent variables changes abruptly at certain points.

For example, a piecewise function could be used to model the cost of shipping a package, where the cost is different for different weight ranges. The function would be defined by one expression for weights below a certain threshold, and a different expression for weights above that threshold.

Notation for Piecewise Functions, Homework 3.6 piecewise functions answers

Piecewise functions are typically written using the following notation:

“` f(x) = g(x) if x< a h(x) if a ≤ x < b k(x) if x ≥ b ```

In this notation, the function f(x) is defined by the expression g(x) for values of x less than a, by the expression h(x) for values of x between a and b, and by the expression k(x) for values of x greater than or equal to b.

Homework 3.6 Answers

Problem	Solution	Graph
Find the value of f(2) for the piecewise function:“`f(x) = x + 1 if x < 0 2x - 1 if x ≥ 0 ```	f(2) = 2(2) 1 = 3
Graph the piecewise function:“`f(x) = -x if x < -1 x + 2 if -1 ≤ x < 1 2x - 1 if x ≥ 1 ```
Find the domain and range of the piecewise function:“`f(x) = |x| if x ≤ 0 x^2 if x > 0“`	Domain: (-∞, ∞)Range: [0, ∞)

Applications of Piecewise Functions

Piecewise functions have a wide range of applications in different fields, including:

• Economics: Piecewise functions can be used to model the demand curve for a product, where the demand is different at different price points.
• Computer science: Piecewise functions can be used to define the behavior of a computer program, where the program executes different code depending on the input.
• Physics: Piecewise functions can be used to model the motion of an object, where the object moves at different speeds or directions at different times.

Examples of Piecewise Functions

• Absolute value function: The absolute value function is a piecewise function that is defined by the following expressions:

“` f(x) =

-x if x< 0 x if x ≥ 0 ```

• Signum function: The signum function is a piecewise function that is defined by the following expressions:

“` f(x) =

-1 if x < 0 0 if x = 0 1 if x > 0

“`

• Heaviside step function: The Heaviside step function is a piecewise function that is defined by the following expressions:

“` f(x) = 0 if x< 0 1 if x ≥ 0 ```

Graphing Piecewise Functions: Homework 3.6 Piecewise Functions Answers

To graph a piecewise function, follow these steps:

1. Identify the different intervals of the independent variable.
2. For each interval, determine the corresponding expression that defines the function.
3. Graph each expression over its corresponding interval.
4. Connect the graphs of the different expressions at the endpoints of the intervals.

Helpful Answers

What is the general notation for a piecewise function?

Piecewise functions are typically expressed using the following notation: f(x) = f1(x), if x< a; f2(x), if a ≤ x < b; ...; fn(x), if x ≥ c, where f1(x), f2(x), ..., fn(x) are different functions defined over different intervals.

How are piecewise functions used in real-world applications?

Piecewise functions find applications in various fields, including economics (modeling piecewise linear demand curves), physics (describing piecewise constant velocity), and computer science (defining piecewise continuous functions for image processing).

What are the advantages of using piecewise functions?

Piecewise functions offer several advantages, including their ability to represent discontinuous relationships, simplify complex expressions, and model real-world scenarios with varying conditions.